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ON PERIODIC WATER WAVES WITH CORIOLIS EFFECTS AND ISOBARIC STREAMLINESANCA-VOICHITA MATIOC a & BOGDAN-VASILE MATIOC aa Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, AustriaPublished online: 04 Mar 2013.

To cite this article: ANCA-VOICHITA MATIOC & BOGDAN-VASILE MATIOC (2012) ON PERIODIC WATER WAVES WITH CORIOLIS EFFECTS AND ISOBARIC STREAMLINES, Journal ofNonlinear Mathematical Physics, 19:sup1, 89-103, DOI: 10.1142/S1402925112400098

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November 23, 2012 11:28 WSPC/1402-9251 259-JNMP 1240009

Article

Journal of Nonlinear Mathematical Physics, Vol. 19, Suppl. 1 (2012) 1240009 (15 pages)

c© A.-V. Matioc and B.-V. MatiocDOI: 10.1142/S1402925112400098

ON PERIODIC WATER WAVES WITH CORIOLIS EFFECTSAND ISOBARIC STREAMLINES

ANCA-VOICHITA MATIOC∗ and BOGDAN-VASILE MATIOC†

Faculty of Mathematics, University of ViennaNordbergstraße 15, 1090 Vienna, Austria

∗anca.matioc@univie.ac.at†bogdan-vasile.matioc@univie.ac.at

Received 19 March 2012Accepted 20 April 2012

Published 28 November 2012

In this paper we prove that solutions of the f -plane approximation for equatorial geophysical deepwater waves, which have the property that the pressure is constant along the streamlines and donot possess stagnation points, are Gerstner-type waves. Furthermore, for waves traveling over a flatbed, we prove that there are only laminar flow solutions with these properties.

Keywords: Periodic water waves; Gerstner’s wave; Coriolis effects; Lagrangian coordinates.

Mathematics Subject Classification 2010: Primary: 76B15, Secondary: 37N10

1. Introduction

The motion of a fluid layer located on the Earth’s surface is also influenced by Earth’srotation around the polar axis. For fluid motions localized near the Equator, the variationof the Coriolis parameter may be neglected and geophysical water waves in this regionare modeled by the so-called f -plane approximation. The physical relevance of the f -planeapproximation for geophysical waves is discussed in [5].

In this paper we consider periodic solutions of the f -plane approximation which possessisobaric streamlines, that is the pressure is constant along the streamlines of the flow.Furthermore, we assume that the wave speed exceeds the horizontal velocity of all particlesin the fluid so that stagnation points are excluded. Based upon a priori properties for suchwaves, which we establish herein, and regularity results for quasilinear elliptic equations, cf.[27], we prove that, if the ocean depth is infinite, such waves have an explicit Lagrangiandescription. These solutions were found initially by Gerstner in [14] and rediscovered lateron by Rankine [29] in the context of flows without Coriolis effects. More recently, theirproperties have been analyzed in [2, 4, 17]. They may be adapted to describe edge wavesin homogeneous [1] and stratified fluids [30, 31], or gravity waves solutions of the f -planeapproximation [23].

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An important characteristic of Gerstner’s solutions is that they describe rotational flowswith the fluid particles moving on circles, a feature which does not hold for irrotationalperiodic deep water waves as seen in [6, 22] by means of linear theory. For the descriptionof the particle paths in linear waves traveling over a flat bottom we refer to [12, 19, 21]. Weenhance that the results on the particle trajectories for linear water waves have nonlinearcounterparts for the governing equations, cf. [3, 11, 16, 18].

In the context of waves in water over a flat bed, we prove that any solution of the f -planeapproximation which has isobaric streamlines and no stagnation points must be a laminarflow, that is the streamlines are straight lines.

Though the angular speed of Earth’s rotation is a well determined constant, our analysisremains valid for any arbitrary value of the angular speed. Particularly, we recover previousresults established for gravity water waves without Coriolis effects [20], which are a specialcase of the situation analyzed herein (zero angular speed). It is worth mentioning that thesymmetry of the Gerstner-type waves reflects the one obtained for gravity waves in waterof finite or infinite depth, cf. [7–9, 26, 28].

The outline of the paper is as follows: we present in Sec. 2 the mathematical formulationof the problem we deal with and state the main results Theorems 2.1 and 2.2. To this endwe introduce in a Lagrangian framework Gerstner solutions for the deep-water problem anddiscuss their properties. In Sec. 3 we reformulate the problem and establish some preliminaryproperties common for both deep and finite depth waves with isobaric streamlines andwithout stagnation points. Section 4 is dedicated to the proof of Theorem 2.1, while inSec. 5 we prove our second main result Theorem 2.2.

2. The Mathematical Model and the Main Results

We consider herein a rotating frame with the origin at a point on Earth’s surface, theX-axis being chosen horizontally due east, the Y -axis horizontally due north, and the Z-axis pointing upward. Furthermore, we let Z = η(t,X, Y ) be the upper free boundary of afluid layer which may have a finite depth, the plane Z = −d, d ∈ R, being the impermeablebottom of ocean, or is unbounded and in this situation we deal with deep water waves. In thefluid layer located near the Equator, the governing equations in the f -plane approximationare, cf. [13], the Euler equations

ut + uuX + vuY + wuZ + 2ωw = −PX/ρ,vt + uvX + vvY + wvZ = −PY /ρ,wt + uwX + vwY +wwZ − 2ωu = −PZ/ρ− g

(2.1)

and the conservation of mass equation

uX + vY + wZ = 0. (2.2)

Here t is the time variable, (u, v, w) the fluid velocity, ω = 73 · 10−6 rad/s is the rotationspeed of the Earth round the polar axis towards east, ρ is the density constant of the water,g = 9, 8 m/s2 is the gravitational acceleration at the Earth’s surface, and P is the pressure.

At the wave surface, the pressure of the fluid matches the atmospheric pressure P0

P = P0 on Z = η(t,X, Y ). (2.3)

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Moreover, the free surface of the wave consists at each moment of the same fluid particles,so that we obtain the kinematic boundary condition

w = ηt + uηX on Z = η(t,X, Y ). (2.4)

Since for finite depth waves the bottom of the ocean is assumed to be impermeable, weimpose the no-flux condition

w = 0 on Z = −d. (2.5)

For deep water waves we assume that

(u,w) → 0 for Z → −∞ uniformly in (t,X), (2.6)

meaning that at great depths there is practically no flow.In this paper we consider traveling waves, with the velocity field, the pressure and

the free surface exhibiting an (t,X)-dependence of the form (X − ct), where c > 0 is thespeed of the wave surface. Moreover, we seek two-dimensional flows, independent upon theY -coordinate and with v ≡ 0 throughout the flow. Introducing the new variables

x := X − ct and z := Z, (2.7)

the governing equations for water waves reduce to the following nonlinear free-boundaryproblem

(u− c)ux + 2ωw = −Px/ρ in Ωη,

(u− c)wx − 2ωu = −Pz/ρ− g in Ωη,

ux + wz = 0 in Ωη,

P = P0 on z = η(x),

w = (u− c)ηx on z = η(x),

(2.8)

supplemented by the boundary condition

w = 0 on z = −d (2.9)

for waves traveling over a flat bed, respectively

(u,w) → 0 for z → −∞ uniformly in x (2.10)

in the infinite-depth case. The fluid domain Ωη is bounded from above by the graph of ηand is unbounded from below in the infinite depth case, respectively is bounded from belowby the line z = −d when the ocean depth is finite.

The solutions we consider are periodic in the variable x, that is u,w, P, η are all periodicin x with the same period, and have no stagnation points. The latter property is satisfiedif we assume that

supΩη

u < c. (2.11)

Moreover, we require a priori that the solutions have the following regularity

η ∈ C3(R) and (u,w, P ) ∈ (C2(Ωη))3. (2.12)

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Our first main result is the following theorem.

Theorem 2.1. Assume that (u, v, P, η) is a solution of (2.8)–(2.9) which satisfies therelations (2.11)–(2.12). If the pressure is constant along the streamlines, then the flow islaminar.

Besides laminar flow solutions, which are characterized by the fact that η is constantand (u, v, P ) depend only upon the z coordinate, in the infinite depth case there is afamily of explicit traveling wave solutions of the problem (2.8) and (2.10), which is due toGerstner. To present these special solutions we adopt a Lagrangian framework and describethe trajectories of each particle in the fluid

(X(t, a, b), Z(t, a, b)) :=(a− ekb

ksin(k(a− ct)), h0 + b+

ekb

kcos(k(a− ct))

)(2.13)

for all a ∈ R, b ≤ b0, and t ≥ 0. Hereby k > 0, b0 ≤ 0, h0 ∈ R is arbitrary, and c isthe speed of the wave. Each particle within the fluid is uniquely determined by a pair(a, b) ∈ R× (−∞, b0). The curve parametrized by (X(t, ·, b0), Z(t, ·, b0)) is the profile of thewave and is a trochoid when b0 < 0, respectively a cycloid when b0 = 0 (see the figures in[4, Chap. 4]). The latter curve has upward cusps, so that the waves cannot be extended forvalues of b ≥ 0. The speed c of the wave is given by

c =−ω +

√ω2 + gk

k,

cf. [23]. Our second main result is the following theorem.

Theorem 2.2. Assume that (u, v, P, η) is a solution of (2.8) and (2.10) which satisfies(2.11)–(2.12). If the pressure is constant along each streamline, then (u, v, P, η) is one ofthe solutions described by (2.13).

3. Equivalent Formulations and a priori Properties of Solutions

In order to investigate the steady flow problems for finite depth and deep water waves wefind equivalent formulations which are more suitable to handle. First, we reformulate theproblem in terms of a stream function ψ. To deal with both finite and infinite depth casesat once, we define the stream function by the following relation

ψ(x, z) := −∫ η(x)

z(u(x, s) − c)ds for (x, z) ∈ Ωη.

Then, it follows by direct computations that ∇ψ = (−w, u− c). Consequently, the stream-lines of the steady flow, which coincide with the particle paths, are the level curvesof ψ. Indeed, if the curve (x(t), z(t)) describes the motion of a fluid particle, that is(x′(t), z′(t)) = (u(x(t), z(t)) − c, w(x(t), z(t))), then

d

dtψ(x(t), z(t)) = 〈∇ψ(x(t), z(t)) | (u(x(t), z(t)) − c, w(x(t), z(t)))〉 = 0 for all t ≥ 0.

Furthermore, using the implicit function theorem and (2.11) we see that the level curves ofψ are in fact graphs of periodic functions defined on the entire real line. Additionally, we

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compute that

∆ψ = ψxx + ψzz = uz − wx = γ in Ωη,

the function γ = γ(x, z) being the vorticity of the flow. By (2.9), for waves over a flat bedψx = −w = 0 on x = −d, and we deduce that there is a positive constant p0 such thatψ = p0 on z = −d. Also, observe that ψ = 0 on z = η(x).

Condition (2.11) enables us also to introduce new variables by means of the hodographtransformation H : Ωη → Ω, defined by

H(x, z) := (q, p)(x, z) := (x,−ψ(x, z)), (x, z) ∈ Ωη,

whereby Ω := (q, p) : −p0 < p < 0 and Ω := (q, p) : p < 0 for the finite and infinitedepth case, respectively. The mapping H is a diffeomorphism and the following relationsare satisfied(

qx qzpx pz

)=

(1 0w c− u

)and

(xq xpzq zp

)=

1 0w

u− c

1c− u

.

A simple computation shows now that ∂q(γ H−1) = 0 in Ω, meaning that there existsa continuously differentiable function γ = γ(p) such that γ(x, z) = γ(−ψ(x, z)) for all(x, z) ∈ Ωη. Finally, defining the hydraulic head by the expression

E :=(u− c)2 + w2

2+ (g − 2ωc)z +

P

ρ− 2ωψ +

∫ −ψ

0γ(s)ds in Ωη,

one can easily show that there exists a constant C such that E = C in Ωη. Introducing theprimitive Γ of γ by the relation

Γ(p) :=∫ p

0γ(s)ds− C,

we find that (η, ψ) solves the following problem:∆ψ = γ(−ψ) in Ωη,

|∇ψ|2/2 + (g − 2ωc)z + P/ρ− 2ωψ + Γ = 0 in Ωη,(3.1)

supplemented by

ψ = 0 on z = −d (3.2)

for waves traveling over a flat bed, respectively

∇ψ → (0,−c) for z → −∞ uniformly in x (3.3)

for deep water waves. In fact, it is not difficult to show that problems (2.8)–(2.9) and(3.1)–(3.2) (respectively, (2.8), (2.10) and (3.1), (3.3)) are equivalent in the sense thateach solution of the first problem corresponds to a unique solution of the second one.Since we consider only waves having the property that the pressure is constant alongthe streamlines we find a function P ∈ C2([−p0, 0]) in the finite depth case (respectively,

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P ∈ C2((−∞, 0]) in the infinite depth case) with the property that P (x, z) = P (−ψ(x, z))for all (x, z) ∈ Ωη.

To obtain a second equivalent formulation of the original problems, we introduce theheight function h : Ω → R by the relation

h(q, p) = z for (q, p) ∈ Ω.

It follows readily from the definition of h and of the coordinates transformation H that hsolves the following equations

(1 + h2q)hpp − 2hphqhpq + h2

phqq − Γ′h3p = 0 in Ω,

1 + h2q

2h2p

+ (g − 2ωc)h +P

ρ+ 2ωp+ Γ = 0 in Ω

(3.4)

and

h = −d on p = −p0 (3.5)

for waves traveling over a flat bed, respectively

∇h→ (0, 1/c) on p→ −∞ uniformly in q, (3.6)

in the infinite depth case. The problems (3.4)–(3.5) and (3.1)–(3.2) (respectively,(3.4), (3.6) and (3.1), (3.3)) are equivalent in the same sense defined before. We notethat (2.11) becomes

infΩhp > 0, (3.7)

while, due to the fact that ψ(q, h(q, p)) = −p for (q, p) ∈ Ω, we see that the streamlines ofthe steady flow are parametrized by the functions h(·, p).

To simplify our notations, we set

α := g − 2ωc and Q(p) :=P (p)ρ

+ 2ωp for all p with (0, p) ∈ Ω. (3.8)

We establish now some properties which are a priori satisfied by the solutions of thesystem (2.8).

Lemma 3.1. Let (u,w, η, P ) be a periodic solution of (2.8) satisfying (2.11)–(2.12), andassume that the pressure P is constant along the streamlines. Then, there exists a con-tinuously differentiable function β = β(p) such that the corresponding height function h

satisfies

1hp(q, p)

= Q′(p)h(q, p) + β(p) for all (q, p) ∈ Ω. (3.9)

Proof. First, we note that the height function h solves (3.4). Differentiating the secondequation of (3.4) with respect to p and q, respectively, we obtain in Ω the following relations

hphqhpq − (1 + h2q)hpp

h3p

+ αhp +Q′ + Γ′ = 0 (3.10)

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and

hphqhqq − (1 + h2q)hpq

h3p

+ αhq = 0. (3.11)

We build the sum of (3.10) and the first equation in (3.4) to find that

h2phqq − hphqhpq

h3p

+ αhp +Q′ = 0 in Ω. (3.12)

Next, we multiply (3.12) by hq, (3.11) by hp and subtracting these new identities we get

hpqh2p

+Q′hq = 0,

or equivalently

∂

∂q

(1hp

−Q′hq)

= 0.

This yields the desired assertion (3.9).

As a further result we prove the following lemma, which enables us later on to identifythe characteristics of the flow.

Lemma 3.2. Under the same assumptions as in Lemma 3.1, the height function h, corre-sponding to a solution of (2.8), satisfies

a3(p)h3 + a2(p)h2 + a1(p)h+ a0(p) = 0 in Ω, (3.13)

whereby

a0(p) := −αβ + 2ββ′(Q+ Γ) −Q′β2 − β2(Q′ + Γ′),a1(p) := −αQ′ − 2Q′β(Q′ + Γ′) + 2(Q+ Γ)(Q′′β +Q′β′),−2Q′2β + 2αββ′,a2(p) := −Q′3 − (Q′ + Γ′)Q′2 + 2α(Q′′β +Q′β′] + 2Q′Q′′(Q+ Γ),

a3(p) := 2αQ′Q′′

(3.14)

for all p with (0, p) ∈ Ω.

Proof. We differentiate first (3.9) with respect to q and multiply the relation we obtain byhq to find that

hqhpq = −Q′h2qh

2p =

Q′

(Q′h+ β)2+

2(αh +Q+ Γ)Q′

(Q′h+ β)4in Ω,

the last identity being a consequence of the second relation in (3.4) and (3.9). On the otherhand, the latter relations yield

h2q = −2

αh+Q+ Γ(Q′h+ β)2

− 1

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and differentiating this expression with respect to p we arrive at

hqhpq = −α+ (Q′ + Γ′)(Q′h+ β)(Q′h+ β)3

+ 2αh+Q+ Γ(Q′h+ β)4

[Q′ + (Q′′h+ β′)(Q′h+ β)].

Identifying the two expressions we found for hqhpq, we obtain the desired relations.

We are interested here to determine the solutions of (2.8)–(2.9) (respectively, (2.8) and(2.10)) which are not laminar (that is hq does not vanish in Ω). For these solutions weobtain the following restriction on the wave speed c.

Lemma 3.3. Assume that (u,w, η, P ) is a periodic solution of (2.8) satisfying (2.11)–(2.12). Furthermore, we assume that the pressure P is constant along the streamlines andthe flow is not laminar, that is hq ≡ 0 in Ω. Then, we have that α = g − 2ωc = 0.

Proof. Let us assume by contradiction that α = 0. Then, the second relation of (3.4) isequivalent to

(1 + h2q)(Q

′h+ β)2 + 2(Q+ Γ) = 0 in Ω.

We differentiate this relation with respect to q and, since Q′h+β > 0, cf. (3.7), we arrive at

hqhqq(Q′h+ β) + hq(1 + h2q)Q

′ = 0 in Ω.

We fix now p < 0 such that h(·, p) is not a constant function. Since h(·, p) is a real analyticfunction, cf. [10, 24], we find that

hqq(q, p) = −Q′(p)1 + h2

q(q, p)Q′(p)h(q, p) + β(p)

for all q ∈ R.

If Q′(p) = 0, then hqq(q, p) = 0 for all q ∈ R, meaning that hq(q, p) = 0 for all q ∈ R andcontradicting our assumption. On the other hand, if Q′(p) = 0, then hqq(·, p) has the signof −Q′(p) on the whole line, in contradiction with the periodicity of hq(·, p0). In conclusion,our assumption was false and the proof is complete.

4. Proof of Theorem 2.1

The proof of Theorem 2.1 follows by contradiction. Assume thus that there exists a tuple(u, v, P, η) which satisfies all the assumptions of Lemma 3.3 and the boundary condition(2.9). We observe that in this case the restriction to the flow of being non-laminar is equiva-lent to saying that η = h(·, 0) is not constant. This follows easily from (3.4)–(3.5) by meansof elliptic maximum principles. Without restricting the generality, we may assume that

hq(·, p) ≡ 0 for all p ∈ (−p0, 0].

Indeed, if this is not the case let

Ω′ = (q, p) ∈ Ω : h(·, p) is not constant.Then, in Ω\Ω′ the flow is laminar and h is constant on the lower boundary of Ω′. Thus,by choosing some other constants p0 and d, we may assume that the height function h

associated to our solution satisfies (3.4) in Ω′ and (3.5) on the lower boundary of Ω′.

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Applying Lemma 3.2 we obtain that

a3 = a2 = a1 = a0 = 0 in [−p0, 0]. (4.1)

We exploit now this relation to arrive at a contradiction. First, from Lemma 3.3 and a3 ≡ 0,we obtain that

Q(p) = Ap +B (4.2)

for all p ∈ [−p0, 0] and with a nonzero constant A and some B ∈ R. Indeed, if A = 0, thenwe infer from (3.9) that h has to be laminar in Ω. Moreover it follows readily from a2 = 0that there exists a constant C0 such that

β(p) =A

2αΓ(p) +

A2

αp+ C0 (4.3)

for all p ∈ [−p0, 0]. Lastly, exploiting the fact that a1 = 0, we identify a differential equationfor Γ

(c0Γ + c1)Γ′ = −(c2Γ + c3) (4.4)

in [−p0, 0], the constants ci being given by the following expressions

c0 :=A2

2α, c1 :=

A2B

α−AC0, c2 :=

A3

α, c3 :=

2A3B

α− αA− 2A2C0. (4.5)

We also need the following result.

Lemma 4.1. We have that(h+

β

A

)2

h2q +

(h+

β

A+

α

A2

)2

= − 1A2

(Γ +

c3c2

)(4.6)

in Ω.

Proof. Invoking the second relation of (3.4), (3.9), and using (4.2) we find that

(Ah + β)2h2q + (Ah+ β)2 + 2αh+ 2Q+ 2Γ = 0

in Ω, or equivalently(h+

β

A

)2

h2q +

(h+

β

A+

α

A2

)2

− 2αβA3

− α2

A4+

2QA2

+2ΓA2

= 0.

The desired relation follows now from (4.3) and (4.5).

Observe from (4.6) that Γ+c3/c2 < 0 for all p ∈ (−p0, 0]. We infer then from (4.4)–(4.5)that there exists a constant C1 such that

Γ +α2

A2ln

(−Γ − c3

c2

)= −2Ap+ C1 (4.7)

in [−p0, 0]. This shows in fact that Γ+c3/c2 < 0 on [−p0, 0]. In turn, this implies c0Γ+c1 = 0for all Γ ∈ [m,M ], where m := min[−p0,0] Γ and M := max[−p0,0] Γ. Defining the functionf : (m − ε,M + ε) → R by f(Γ) := −(c2Γ + c3)/(c0Γ + c1), we see that this functionis well-defined and real-analytic on (m − ε,M + ε) provided ε is small. Since Γ′ = f(Γ),

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we deduce that Γ possesses an extension (which is called also Γ) defined on some interval(−p0− δ, δ) with δ > 0. Additionally, Γ inherits the regularity of f , that is Γ is real-analyticon (−p0 − δ, δ).

We define now the function h : R × [−p0 − δ/2, 0] → R by setting h(q, p) = h(q, p) in Ω,and

h(q, p) := h(q,−p0) +∫ p

−p0

1√1/h2

p(q,−p0) + 2Γ(−p0) − 2Γ(s)ds

for p ∈ [−p0 − δ/2,−p0]. We note first that h is well-defined. Indeed, this is a consequenceof the fact that h(q,−p0) = −d and of relation (3.9). Moreover, h ∈ C2(R × [−p0 − δ/2, 0])is the solution of the quasilinear elliptic equation

(1 + h2q)hpp − 2hphqhpq + h2

phqq − γh3p = 0 in R × [−p0 − δ/2, 0],

with h satisfying (3.7) in R×[−p0−δ/2, 0]. We enhance that condition (3.7) guarantees thatthis quasilinear equation is uniformly elliptic. Since γ is real-analytic and all three equationsof the system depend analytically on h, the regularity results in [15, 27] imply that h isreal-analytic in the strip R × (−p0 − δ/2, 0). But, since hq ≡ 0 in R × [−p0 − δ/2,−p0], theprinciple of analytic continuation implies the flow must be laminar, in contradiction withour assumption. This finishes the proof of Theorem 2.1.

5. Proof of Theorem 2.2

In this section we consider a non-laminar solution (u, v, P, η) which satisfies all the assump-tions of Lemma 3.3 and the far field condition (2.10), that is a solution of the deep waterwave problem. Furthermore, we let h be the corresponding height function. From the proofof Theorem 2.1 we deduce that it is possible to choose p0 ≥ 0 such that hq(·, p) ≡ 0 for allp ≤ −p0.

We shall restrict our attention first to the flow in Ω′ := R× (−∞,−p0). By the results ofSec. 4, the relations (4.2)–(4.4) (and subsequently (4.7)) are satisfied on (−∞,−p0], and theidentity (4.6) takes place in all Ω′. Particularly, Γ + c3/c2 < 0 in (−∞,−p0]. This impliesthat

A < 0. (5.1)

Indeed, if A > 0, then we infer from (4.7) that Γ(p) → ∞ as p → −∞, which contradictsthe fact that Γ is bounded from above.

Our previous analysis allows us to prove now that the flow possesses some of the char-acteristics of Gerstner’s solution for deep water waves in the f -plane approximation.

Lemma 5.1 (The flow is rotational). There exists ε > 0 such that γ is real-analytic on(−∞,−p0 + ε). Moreover, γ′ < 0 and γ →p→−∞ 0.

Proof. Since Γ < −c3/c2 in (−∞,−p0], we infer from Eq. (4.4) that Γ(p) = −c1/c0 forall p ∈ (−∞,−p0]. It follows readily from (4.5) that −c1/c0 < −c3/c2, so that eitherΓ < −c1/c0 or −c1/c0 < Γ < −c3/c2. By (4.7) and (5.1), we can exclude the alternative

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Γ < −c1/c0, so that

− c1/c0 < Γ < −c3/c2 for all p ∈ (−∞,−p0]. (5.2)

Particularly, the same arguments as in Sec. 4 imply that Γ is real-analytic on (−∞,−p0+ε),provided ε > 0 is small, and since by (4.4)

γ = Γ′ = −c2c0

Γ + c3/c2Γ + c1/c0

= −2AΓ + c3/c2Γ + c1/c0

,

we use (5.1) and (5.2) to obtain that γ < 0 and γ →p→−∞ 0 (or equivalently Γ(p) →p→−∞−c3/c2). Differentiating the last equation once we see that γ′ < 0. This completes ourargument.

Since γ is real-analytic on (−∞,−p0 + ε), the regularity results in [25] imply that allthe streamlines h(·, p), p < −p0 +ε, are real-analytic functions. Even more, using regularityresults for quasilinear elliptic equations [27], we know by (3.7) that h is real-analytic in Ω′.Let now

K(p) := − 1A

(−Γ − c3

c2

)1/2

for p ≤ −p0.

With this notation, Eq. (4.6) becomes(h+

β

A

)2

h2q +

(h+

β

A+

α

A2

)2

= K2 in Ω′. (5.3)

Furthermore, we define

(x(s, p), z(s, p)) :=(α

As−K(p) sin(As),− α

A2− β(p)

A+K(p) cos(As)

)(5.4)

for all s ∈ R and p ≤ −p0. Fixing p ≤ −p0, we have that

xs =α

A−AK(p) cos(As), s ∈ R

and since K(p) →p→−∞ 0 we may take p0 large to guarantee that x(·, p) : R → R is adiffeomorphism for all p ≤ −p0. This fact allows us to define the map s : R×(−∞,−p0] → R

by the relation x(s(q, p), p) = q for all (q, p) ∈ R × (−∞,−p0]. Note that the implicitfunction theorem ensures that s is real-analytic as well. Finally, let h : R× (−∞,−p0] → R

be defined by

h(q, p) := z(s(q, p), p) for (q, p) ∈ R × (−∞,−p0].

Then, using the chain rule, we find that h satisfies the following relation(h+

β

A

)2

h2q +

(h+

β

A+

α

A2

)2

=(− α

A2+K cos(As)

)2 (zsxs

)2

+K2 sin2(As)

=(α

A2−K cos(As)

)2 A2K2 cos2(As)(α/A−AK cos(As))2

+K2 sin2(As)

= K2.

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Since h(·, p) and h(·, p) are both real-analytic solutions of (5.3), by the theorem of Picard–Lindelof we find a function δ = δ(p), p ≤ −p0, such that

h(q, p) = h(q + δ(p), p) in R × (−∞,−p0]. (5.5)

Using the implicit function theorem and (4.6), we may choose δ to be a real-analytic func-tion. Furthermore, since h(q, p) → −∞ when p→ −∞, we infer from (4.3), (5.4) and (5.5)that

α > 0. (5.6)

Given s ∈ R and p ≤ −p0, we let q ∈ R satisfy s = s(q+δ(p), p). Then q = x(s, p)−δ(p),h(q, p) = h(q + δ(p), p) = z(s(q + δ(p), p), p) = z(s, p)

and therefore we have

(u− c)(x(s, p) − δ(p), z(s, p)) = (u− c)(q, h(q, p)) = − 1hp

(q, p) = −Ah(q, p) − β(p)

= −Az(s, p) − β(p) =α

A−AK(p) cos(As)

=∂

∂s(x(s, p) − δ(p)),

respectively

w(x(s, p) − δ(p), z(s, p)) = w(q, h(q, p)) = −hqhp

(q, p) = −h(q + δ(p), p)(Az(s, p) + β(p))

=−AK(p) sin(As)

αA −AK(p) cos(As)

(αA

−AK(p) cos(As))

=∂z

∂s(s, p).

This shows that the path of an arbitrary particle in the steady flow beneath y = h(·,−p0)is described by the curve

t → (x(t+ s, p), z(t+ s, p)) := (x(t+ s, p) − δ(p), z(t + s, p)),

whereby (x(s, p), z(s, p)), with (s, p) ∈ R × (−∞,−p0] is the initial position of the particle.Back to the original reference frame (X,Z), the position of the particles at any time isdescribed by the mapping

(t, s, p) → (X(t, s, p), Z(t, s, p)) := (x(t+ s, p) + ct, z(t+ s, p))

for (t, s, p) ∈ [0,∞)×R× (−∞,−p0]. Indeed, one can easily check that (X(·, s, p), Z(·, s, p))are solutions of the non-autonomous system

Xt = u(t,X,Z) = u(X − ct, Z),

Zt = w(t,X,Z) = w(X − ct, Z)

for each s ∈ R and p ≤ −p0.

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In the remainder of this section we show that the equations for the particle paths canbe brought in the form (2.13). To this end, we introduce new variables

a :=α

As and b := T (p) =

ln(A2

α K(p))

A2

α

,

with a ∈ R and b ≤ b0 := H(−p0). That the map T : (−∞,−p0] → (−∞, b0] is a diffeomor-phism is a consequence of the relation

T ′(p) =α

A2

K ′(p)K(p)

=α

2A2

2K(p)K ′(p)K2(p)

= − α

2A4

γ(p)K2(p)

> 0 for p ≤ −p0,

cf. (5.6) and Lemma 5.1. Letting the positive constants k and m be defined by

k :=A2

αand m := −α

A,

the paths of the particles located initially beneath the curve (X(0, ·,−p0), Z(0, ·,−p0)),which we may regard as being the surface of the wave, are described in the reference framelocated on the Earth’s surface by the mappings

X(t, a, b) := −δ(b) + a+ (c−m)t− ekb

ksin(k(a −mt)),

Z(t, a, b) := h0 + b+ekb

kcos(k(a−mt)),

(5.7)

with a ∈ R, b ≤ b0. Hereby, we have defined δ := δ T−1 and h0 is a suitable constant. Thesecond equation of (5.7) follows from the relation

(β T−1)′(b) =A2

α

(β′K

K ′

)H−1 =

2A2

α

(β′

K2

(K2)′

)H−1 = −A for all b ≤ b0,

which is a consequence of the fact that (4.7) may be written in the equivalent form

β = − α

2Aln(A2K2) +

AC1

2α− C0 in (−∞,−p0].

That the flow generated by (5.7) is incompressible reduces to showing that

XtaZb −XtbZa − ZtaXb + ZtbXa = 0 in [0,∞) × R × (−∞,−p0],

cf. [23, Lemma 3.4]. However, the latter condition is equivalent to δ′ = 0, meaning thatδ is a constant function. Translating the original solution h suitably, we may assumewithout restricting the generality that δ ≡ 0. Back to the coordinates (s, p) we havethat

(u− c)(x(s, p), z(s, p)) =α

A−AK(p) sin(As) → α

A,

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A.-V. Matioc & B.-V. Matioc

when p → −∞, uniformly in s. But then, by (2.10), we identify m with the wave speedm = c, so that the trajectories of the particles are described by

X(t, a, b) = a− ekb

ksin(k(a− ct)),

Z(t, a, b) = h0 + b+ekb

kcos(k(a− ct))

(5.8)

for all a ∈ R b ≤ b0 and t ≥ 0.Since h is real-analytic in the interior of the set (q, p) : hq(·, p) ≡ 0, we conclude that

Ω′ = Ω, and that there exists a b0 ≤ 0 such that our original solution is described by theEqs. (5.8) in Ωη. This finishes the proof.

Acknowledgment

A.-V. Matioc was supported by the FWF Project I544–N13 “Lagrangian kinematics ofwater waves” of the Austrian Science Fund.

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